approximation of the log function

Because

limx→ 0⁡x⁢log⁡(x)

=

limx→0⁡xx-1

we can approximate log⁡(x) for small x:

log⁡(x)

≈

xx-1x.

A perhaps more interesting and useful result is that for x small we have the approximation

log⁡(1+x)≈x.

In general, if x is smaller than 0.1 our approximation is practical. This occurs because for small x, the area under the curve (which is what log is a measurement of) is approximately that of a rectangle of height 1 and width x.

Now when we combine this approximation with the formulalog⁡(a⁢b)=log⁡(a)+log⁡(b), we can now approximate the logarithm of many positive numbers. In fact, scientific calculators use a (somewhat more precise) version of the same procedure.

For example, suppose we wanted log⁡(1.21). If we estimate log⁡(1.1)+log⁡(1.1) by taking 0.1+0.1=0.2, we would be pretty close.